I read a book to my pet and it said that it gained an alchemy point. What does that mean? I've never seen this before.

It means your pet reached a multiple of 16 in intelligence! For every 16 points of intelligence you get 1 alchemy point. You have to have a certain number of alchemy points to teach your pet spells that you create at the secluded shrine.

Alchemy Points (AP) Are used for Creating and Equipping Scrolls, which help you in the Battle Coliseum, They are obtained from Reading books, And Mystic Spindles upon the 16th Multiple of Intelligence .

More information on Scrolls/Spells/AP can be found Here

Thanks for your help!